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Local polynomial estimation of regression operators from functional data with correlated errors

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  • Benhenni, Karim
  • Hassan, Ali Hajj
  • Su, Yingcai

Abstract

This article considers the problem of nonparametric estimation of the regression operator r in a functional regression model Y=r(x)+ε with a scalar response Y, a functional explanatory variable x, and a second order stationary error process ε. We construct a local polynomial estimator of r together with its Fréchet derivatives from functional data with correlated errors. The convergence in mean squared error of the constructed estimator is studied for both short and long range dependent error processes. Simulation studies on the performance of the proposed estimator are conducted, and applications to independent data and El Niño time series data are given.

Suggested Citation

  • Benhenni, Karim & Hassan, Ali Hajj & Su, Yingcai, 2019. "Local polynomial estimation of regression operators from functional data with correlated errors," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 80-94.
    • Handle: RePEc:eee:jmvana:v:170:y:2019:i:c:p:80-94
    DOI: 10.1016/j.jmva.2018.10.008
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    References listed on IDEAS

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    1. Rachdi, Mustapha & Laksaci, Ali & Demongeot, Jacques & Abdali, Abdel & Madani, Fethi, 2014. "Theoretical and practical aspects of the quadratic error in the local linear estimation of the conditional density for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 73(C), pages 53-68.
    2. Aldo Goia & Philippe Vieu, 2015. "A partitioned Single Functional Index Model," Computational Statistics, Springer, vol. 30(3), pages 673-692, September.
    3. Karim Benhenni & Sonia Hedli-Griche & Mustapha Rachdi, 2017. "Regression models with correlated errors based on functional random design," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 1-21, March.
    4. A. Berlinet & A. Elamine & A. Mas, 2011. "Local linear regression for functional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(5), pages 1047-1075, October.
    5. Germán Aneiros & Philippe Vieu, 2015. "Partial linear modelling with multi-functional covariates," Computational Statistics, Springer, vol. 30(3), pages 647-671, September.
    6. K. Benhenni & F. Ferraty & M. Rachdi & P. Vieu, 2007. "Local smoothing regression with functional data," Computational Statistics, Springer, vol. 22(3), pages 353-369, September.
    7. He, Guozhong & Müller, Hans-Georg & Wang, Jane-Ling, 2003. "Functional canonical analysis for square integrable stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 54-77, April.
    8. Kara, Lydia-Zaitri & Laksaci, Ali & Rachdi, Mustapha & Vieu, Philippe, 2017. "Data-driven kNN estimation in nonparametric functional data analysis," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 176-188.
    9. A. Pérez-González & J. Vilar-Fernández & W. González-Manteiga, 2009. "Asymptotic properties of local polynomial regression with missing data and correlated errors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(1), pages 85-109, March.
    10. Aneiros-Pérez, Germán & Vieu, Philippe, 2008. "Nonparametric time series prediction: A semi-functional partial linear modeling," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 834-857, May.
    11. Benhenni, K. & Hedli-Griche, S. & Rachdi, M., 2010. "Estimation of the regression operator from functional fixed-design with correlated errors," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 476-490, February.
    12. Chagny, Gaëlle & Roche, Angelina, 2016. "Adaptive estimation in the functional nonparametric regression model," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 105-118.
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    Cited by:

    1. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.
    2. Karim Benhenni & Ali Hajj Hassan & Yingcai Su, 2024. "The effect of correlated errors on the performance of local linear estimation of regression function based on random functional design," Statistical Papers, Springer, vol. 65(6), pages 3395-3423, August.

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